Density-Robust Spherical Coordinates from Persistent Cohomology

2026-07-27

AI Summary

This work addresses the severe distortion of sphere-valued coordinates derived from persistent cohomology under non-uniform sampling, which hinders their practical applicability. To overcome this limitation, the authors propose a robust framework for constructing spherical coordinates that is resilient to sampling density variations. The approach first obtains an approximately uniform subsample via rejection sampling and computes local spherical coordinates, then integrates them into a globally consistent map using spherical Procrustes alignment. Key innovations include the first method achieving robustness to non-uniform sampling, a computationally tractable Euclidean relaxation for efficient spherical alignment, and a memory-efficient implementation that surpasses the scalability limits of classical techniques. Experiments demonstrate that the method accurately recovers spherical structure even under strong sampling bias, scales to datasets with tens of thousands of points, and maintains high precision under uniform sampling conditions.

Abstract

Persistent cohomology provides a principled framework for constructing nonlinear coordinates that reflect the topology of data. However, these topological coordinates can be severely distorted by non-uniform sampling density, limiting their applicability to real-world data. While density-robust circular coordinates have recently been developed, the extension to spherical coordinates remains an open challenge: unlike the circular case, spherical coordinates are obtained through a nonlinear variational problem for sphere-valued maps, to which existing density-correction mechanisms are not directly applicable. In this paper, we introduce the first density-robust construction of spherical coordinates from persistent cohomology. Rather than modifying the coordinate optimization itself, we extend a subsampling-and-alignment framework for circular coordinates to $S^2$, which first computes spherical coordinates on approximately uniform subsamples obtained by rejection sampling and then combines them into a global consensus map. The principal mathematical difficulty is the alignment of independently computed sphere-valued coordinates. We formulate this challenge as a spherical Procrustes problem and establish approximation guarantees for a computationally tractable Euclidean relaxation. Our resulting construction is robust to non-uniform sampling and retains the accuracy of classical spherical coordinates under uniform sampling. Moreover, by computing persistent cohomology only on fixed-size subsamples, our approach avoids the quartic memory bottleneck of the classical spherical coordinate pipeline and scales to substantially larger datasets. We conduct experiments on synthetic data and demonstrate accurate coordinate recovery under severe sampling bias and scalability to datasets of 10,000 points.

Problem

• persistent cohomology
• spherical coordinates
• density robustness
• non-uniform sampling
• topological coordinates

Innovation

• density-robust
• spherical coordinates
• persistent cohomology
• spherical Procrustes problem
• subsampling-and-alignment